ω→0lim= Z ∞
−∞
dvk Z ∞
0
dv⊥v⊥ 1 ω−kkvk
∂
∂r n0 Ti3/2
J02e−v2/2v2T i
!
= lim
ω→0= Z ∞
−∞
dvke−vk2/2v2T i ω−kkvk
∂
∂r n0
Ti3/2 Z ∞
0
dv⊥v⊥e−v2⊥/2v2T iJ02(k⊥ρi)
!
=− π kk
ω→0lim
∂
∂r
"
e−vRes2 /2v2T i n0 T01/2
Γ0(bi)
#
(ρi = v⊥
Ωci, not vT i Ωci !) wherevRes=ω/kk →0,bi≡ky2ρ2T i.
We used Plenelji formula
=
1 ω−kkvk
=− π kk
δ
ω kk
−vk
Therefore, for vRes →0,
∂
∂r
"
n0(r) T01/2(r)
Γ0 k2⊥ρ2i
#
= 0
⇒ The most simple magnitude profile.
Fork⊥ρi→0 (long wavelength limit),
∂
∂r
"
n0(r) T01/2(r)
#
= 0
Note For ∂r∂
n0(r) T01/2(r)
= 0, Ti(r)∝n0(r)2 radially.
⇒ Ti(edge) determines Ti(core) if transport is stiff.
If ITG instability is violent enough, it can throw out ion heat rapidly to outside as soon asTi gets heated (e.g. by NBI) above the threshold condition.
For a known density profile n0(r) and the wavelength of fluctuation k⊥, we can determine the ion temperature profile.
⇒ ITG turbulence is also called “stiff turbulence”.
No matter what the input power is, profile go back to marginal profile!
Of course, it’s most pessimistic scenario.
ITG instability is hard to handle, so it’s one of the reason why people are focused on the improvement at the edge.
0.7< ρ=r/a <0.85: core-edge connection region (No man’s land!) 6.2. Onset Condition of ITG Instability (Nov. 28) The onset condition here refers the linear threshold.
ηi = dlnTi dlnn0 = Ln
LTi ≥ 2
1 + 2bi[1−I1(bi)/I0(bi)] withbi =k⊥2ρ2i in a uniformB field (Kadomtsev and Pogutse).
“eta-i mode”:
For flat density profileLn→ ∞, there’s an instability for very weak ∇Ti.
There should be a threshold condition (forLn→ ∞) in terms of 1/LTi ≥(1/LTi)crit orR0/LTi ≥(R0/LTi)crit. In the long wavelength limit, bi≡(k⊥ρi)2 →0 and the
gyrokinetic equation simplifies to the drift-kinetic equation
∂t+vk b· ∇+vd· ∇+δuE· ∇ − 1
m q∇kδφ+µ∇kB
∂vk
F = 0 where vd· ∇ come from gradient and curvature drifts and µ∇kB is the mirror force. After linearization,
∂t+vk b· ∇+vd· ∇
δf ⇒ −i ω−kkvk−ωdi δf
which is the linear propagator andωdi =k·vd (cf. for Vlasov, we would have an additional Ωci). Therefore, the wave can resonate with particles’ motion along B (vk) and/or particles’ drift motion across B (vd). It’s very difficult to keep both resonances analytically, however each resonance can be handled.
• Sheared slab: ωdi is ignored in ω−kkvk−ωdi
• Simplified toroidal calculation: kkvk is ignored in ω−kkvk−ωdi
Keeping onlyω−kkvkresonance, withkk =kyx/Ls,Ls=qR0/ˆsandx=r−rm/n, q
ˆ s
R0 LTi ≥
Ls LTi
crit
= 1.9 Ti
Te + 1
forLn→ ∞. Favorable role of Ti/Te (hot ion mode).
Keeping onlyω−ωdi resonance, with ωdi=−(cTi/eB) 1/R0, R0
LTi ≥ 4 3
Ti
Te + 1
Likewise, favorable role of Ti/Te.
More complicated formulas exist for finiteLn.
Steve Scott [Phys. Rev. Lett. 29, 531 (1990)]: ηi &ηicrit of above formulas χi ∼χφ (ion thermal diffusivity∼momentum diffusivity)
(NBI plasmas in TFTR, PPPL)
In retrospect, it’s hard to understand why most people were reluctant to accept a possibility thatχφ is anomalous! χneoφ χneoi ! (Mattor and Diamond ’88)
Note Why diamagnetic drift frequency is contained in GK equation, which is for guiding center? ⇒ Information is contained in equilibrium distribution which describe many particles i.e. F0 → n0. In propagator which describes motion of one particle, there’s no diamagnetic drift!
6.3. Basic Properties of ITG Instability in Toroidal Geometry Unstable ITG should have characteristics related to Rayleigh-Taylor instability (contrast to negative compressibility ITG in slab geometry).
⇒ Motivates a “local” theory at bad curvature side (largeR).
Each guiding-center drift:
v∇B+curv=v∇B+curv
v2k+µB vT i2
!
High energy ions drift faster!
0. Seed perturbation inpi
1. vdi∝
v2k+µB
⇒ hot particles drift faster and density build up occurs below hot spots.
2. Ebuilds up in toroidal direction due to ion surplus (deficiency) at high (low)ne 3. radial E×B drift results→ “0.”,
reinforce the initial seed perturbation →instability
6.4. Fluid Description of ITG Instability in Toroidal Geometry (Dec. 3)
Pursue a local theory in the bad curvature region.
In the previous lecture, we’ve learned that energy dependence of∇Band curvature drift led to an instability using figures. This time, we derive a simplified linear dispersion relation.
Linearized drift-kinetic equation (krρi 1):
∂
∂t+vk∇k+vdi· ∇
δf+ c
B∇δφ×b· ∇ − q
m∇kδφ ∂
∂vk
F0'0 where vdi is the gradient and curvature drift. If we take a velocity moment, Rd3v= 2πR
dµdvkB of this, we obtain a continuity equation for δni.
∂
∂tni+δuE· ∇n0+n0∇kδuk+in0
TiωdiδTi+· · ·= 0
where the underlined term is a consequence of the energy dependence ofvdi, i.e., vdi'vdi,th
vk2+µB
/v2th,i →δni couples toδTi!
We can also derive an evolution equation forδpi by takingR
d3v 12mv2
moment of the drift-kinetic equation.
The simplest model for the evolution ofδTi is (note that for linear physics δpi = Tiδni+niδTi), ∂t∂δTi+δuE · ∇Ti '0, i.e., an E×B convection. Then we have
∂
∂tδTi+iω∗Ti|e|δφ'0 with
ωdi≡ − cTi eBR0ky (at bad curvature side), and
ω∗Ti ≡ −cTi
eB ky
LTi
the ion temperature diamagnetic frequency, where we haveω∗e = (cTe/eB) (ky/Ln).
Taking a limit Ln → ∞ and kk → 0 (simplified toroidal calculation), with
δni/n0=δne/n0=|e|δφ/Te (Boltzmann response for electrons, adiabatic),
−iωδni
n0 +iωdiδTi
Ti0 '0
−iωδTi Ti0 +
Te Ti
iω∗Ti
|e|δφ Te '0 2x2 equations for δTi and δφ⇒ det [ ] = 0.
ω2 =−Te Ti
|ωdiω∗Ti| or γ2 = Te Ti
|ωdiω∗Ti| Instability, sinceγ >0 !
This simplified toroidal instability shows a hybrid nature, i.e. we need both
• ion temperature gradient∼ |ω∗Ti|
• bad curvature∼ |ωdi|
A similar example has been considered for MHD description of Rayleigh-Taylor instability.
Can we justify a local consideration at bad curvature side?
→ ignoring good curvature side (high-B side) requiresλk ≤2πqR⇒kk ≥1/qR.
This is incompatible with an approximation used for the derivation,kk →0.
Why did we spend so much time dealing with magnetic geometry and Ballooning Mode Formalism?
In uniformBfield and sheared slab model, we kept kk. We’ve learned that for an instability to exist,kk cannot be too large. Otherwise, the ion Landau damping, shear-induced damping can stabilize the mode.
⇒ kk = 2π/λk should be kept small enough
⇒ Flute-like mode structure: λk >2πqR !
The mode structure needs to be “ballooning at the outside” and “flute-like along B” simultaneously.
Why do we care about the mode structure of instabilities?
Mode width can be regarded as an approximation of the turbulent eddy size.
Turbulent eddy size is related to the anomalous transport rate due to turbulence.
Using a random walk argument, Dturb∼ (∆x)2
∆t ∼ (eddy size)2
eddy life time (circulation time)
If we’re forced to quantities which are available from linear theory,
∆x∼λx ∼k−1x
∆t∼γlin−1 Dturb∼(∆x)2
∆t ∼λ2xγlin∼ γlin k⊥2
at a conceptual level, many transport calculations are more quantitative elabora- tions of this consideration. This is whyλx, γlin of an instability are quantities of practical interest for magnetic fusion.
6.5. Simple Estimation of Turbulent Transport (Dec. 5) Note very rough estimation, could be off by a factor!
If we’re forced to quantities which are available from linear theory, Dturb∼ (∆x)2
∆t ∼ γlin kx2 This is also related to
γeff =γlin−k2xDturb
where the correcting factor is the damping due to nonlinear coupling to other modes (sometimes people putk⊥ instead ofkx).
(Kadomtsev)
Turbulent diffusion increases withλx ∼1/kx and γlin. If we take
γlin∼ω∗∼ kyρivT i
L where
“macroscopic length”L ⇒LT i for ITG
⇒Ln for electron drift instability
⇒LT e for TEM
and so on,
Dturb∼ ky
k2xρi
ρi
L
cTi
eB
Therefore from k-spectra of turbulence, we can guess the scaling of transport.
For instance, ifkx∼ky ∝ρ−1i i.e. λx∼λy ∝ρi, Dturb∼ρi
L
cTi
eB
: gyro(reduced)-Bohm scaling Ifλx∼λy ∝a∼L,
Dturb ∝ cTi
eB
: Bohm scaling
Define ρ∗ = ρi/a, and it’s value is 103 for ITER, 1/300 for JET, 1/100−1/200 for KSTAR/DIII-D/AUG...
Why has Bohm scaling been observed in the early days (∼ 60s, 70s) of magnetic confinement research?
− ~ 2m
∂2
∂x2 +V (x)
ψ=Eψ for SHO in QM,ψ∝ψ`x(x)∝H`x(√
σx)e−σx2/2 (quantization condition).
In wave guides, quantization is based on system size. Same thing there.
There was very smalla, and relatively weakB
⇒ ρi/awas NOT very small (&1/10)
⇒ it’s likely that even drift wave eigenmodes have low quantum number`x
⇒ λx, λy = fraction of system size∝a∝ρi
From experiments, Bohm-like scaling persisted even up to present days, in partic- ular (forion heat transport) in NBI-heated L-mode plasmas.
In those days,TeTi. From experiments, Bohm-like scaling persisted even up to present days, in particular for ion heat transport in NBI-heated L-mode plasmas.
This even thoughρ∗ <10−2 in DIII-D for instance!
Most ITG or drift wave theory for tokamaks lead to gyro-Bohm.
From thek-spectra of turbulence, we can guess the scaling of transport.
In the early 90’s, radially elongated eddys (or global toroidal eigenmodes) were used to explain Bohm-like transport. In toroidal geometry, θ is no longer sym- metric: the poloidal harmonicsm couple with each other!
The radial width of a global toroidal eigenmode is determined by profile variations which break translational invariance. For example, ifω∗Ti(r) varies withL∗:
ω∗Ti(r) =ω∗Ti(r0) 1− (r−r0)2 L2∗
!
⇒ ∆r∝p
L∗ρi while ky ∝ρ−1i Then
k2x∼(ρiL∗)−1 ⇒ Dturb∼ cTi
eB
even for tokamak plasmas.
cf. from magnetic shear in sheared slab model,
∆rDW∼ρsp
Ls/Ln ⇒ still rDW∼ρi 6.6. Zonal flows (Dec. 10)
Dominance of radially elongated eddys, global toroidal eigenmodes, streamers
⇒ high turbulence level and Bohm-like transport.
However, since 90’s, it has been observed from simulations that turbulent eddys get broken up by self-generated (turbulence-generated) zonal flows.
δΦZF=δΦZF(r, t) ; independant of θand φ(ζ) Zonal flow is represented by fluctuation.
uZF = cb× ∇δΦZF B
⇒ binormal (mostly poloidal) direction
→ E×B zonal flows or radial mode (kr 6= 0,kθ=kφ= 0) Note thatuZF,r = 0 !
→ Zonal flows are linearly “stable”.
They cannot tap free energy in (mean) ∂T∂ri and ∂n∂r.
→ They can only be driven nonlinearly.
Reference Review of zonal flows [Diamond et al., PPCF 47, R35 (2005)]
1. Global simulations
density fluctuation contours from ITG turbulence
⇒ lower level for turbulence and transport∼gyroBohm!
2. Flux-tube simulations (quasi-global)until early 90’s
• ω∗Ti(r), ω∗e(r) = constant in radius
• q(r),ˆs= constant in radius
• Radially periodic boundary conditions for turbulence
⇒ Efficient! ∆rturb L ∼a
→ They have found the importance of ZFs∼’93
Until early 90’s: global gyrokinetic simulation, ρ−1∗ . 100,125. The ZFs from these simulations were either in system-size or suppressed!
χGyrofluidi > χGKi ? χFlux tube, GK
i > χGlobal, GKi ? (most flux-tube simulations were gyro-fluid)
1996, J. Glanz Science article:
ITER (based on old design) will fail like the Titanic! Based on (mostly) ITG transport model heavily relying on nonlinear gyrofluid simulations.
1998, Rosenbluth and Hinton, Phys. Rev. Lett. 80, 724 (residual zonal flow):
Gyrofluid equations underestimates zonal flows (overestimates zonal flow damp- ing)⇒ importance of ZFs.
Before this work, most simulation codes were bench-marked for γlin of unsta- ble ITG. After this, this “RH” ZF damping test is widely used as well. From simulations (numerical experiments), one can suppress ZF articicially, or keep it naturally. ⇒ contrast two sets of simulations, to isolate the effects of zonal flows.
Effects of zonal flows on (ITG) turbulence
• Zonal flows reduce “turbulence eddy size”
• Zonal flows reduce turbulence amplitude
• Therefore, zonal flows reduce turbulent transport!
⇒ Paradigm shift: need to incorporate ZFs which regulate ITG turbulence.
Outstanding (frequently asked) questions about zonal flows:
1. How do turbulence eddys get broken up? (beyond movies showing it) 2. How do ZFs get generated?
• For 1, it’s useful to consider the effects of mean E×B shear flows on turbu- lence. (related to H-mode and ITB physics.)
• For the understanding of 2, consideration of “conservation laws” can provide useful insight.
Of course, one can do a non-linear mode-coupling analysis for this, but we ran out of time for this class.
6.7. Duality of Eddy Shearing (due to Zonal Flows) and Sponta- neous Zonal Flow Generation (Dec. 12)
For this two-component system (turbulence + zonal flows), the total energy should be conserved. (Diamond et al. ’95)
Etot=EZF+Eturb = 1 2
X
q
|vZF,q|2+1 2
X
k
Ek
whereq= (qr,0,0) is the wave number for ZFs and Ek= Te
2 X
k
1 +k⊥2ρ2s
|δφk|2
(k2⊥=k2r+k2y) for a drift wave.
ωDW = kyv∗e
1 +k2⊥ρ2s ≡ ω∗e
1 +k⊥2ρ2s ωZFωDW
⇒ adiabatic invariant, i.e. drift-wave “action” densityNk is conserved.
Quasi-particle picture (duality of wave and particle) ⇒Ek=Nkωk. Note that streaming of eddys⇒ ∆r &,kr% therefore ωDW&
⇒ Ek&(∵Nk is conserved) ⇒ ZF energy% (i.e., ZF generation/growth) 6.8. Outstanding Issues of Turbulent Transport in Tokamaks
Qi Qe
Γφ Γp
=−
χi . . . . . . . χe . . . . . . . χφ . . . . . . D
(∇Ti)r (∇Te)r (∇Uφ)r (∇n)r
Generalization of Fick’s law Γp=−D∂n∂x (→ diffusion equation).
Exist off-diagonal pinch terms.
τE % with Ip more strongly than Bp!
Ion Thermal Transport (typically due to ITG contribution)
One candidate: ZF shearing gets ineffective if geodesic acoustic side band (of ZF) withωGAM ∼cs/R0 gets stronger at the expense of the main (ωZF ∼0) ZF.
GAMs can be Landau damped withγdamping ∼ −e−q2/2 where q'rBφ/RBθ.
∴largeq → strong GAM → strong turbulence.
Electron Thermal Transport
No dominant candidate for every case. Depending on parameters.
A. Trapped electron mode (TEM)
A strong evidence from AUG ECH experiments (F. Ryter, PRL ’05) DTEM (dissipative TEM)⇒ Neo Alcator scalingτE ∝neaR2
(’82 for ohmic plasmas), but GK codes find ITG more unstable than TEM!?
B. Electron temperature gradient mode (ETG) Associated ZFs are relatively weak
→ radially elongated eddys
→ streamer could dominate
Otherwise ∆r∼several ρe ⇒transport small Microtearing Mode
Instead ofE×Btransport due to electrostatic fluctuation δvr = cb× ∇δφ
B transport mechanism due to magnetic flutter
δvr= δBr
B0
vk
Momentum Transport
Spontaneous/intrisic rotation of plasmas in the absence of external torque input from NBI, ICRH ... (even for Ohmic plasmas!)
NBI not efficient in driving rotation in ITER.→ rotation of ITER plasma?
(for resistive wall mode, turbulence, etc.)